Question 1192415
**a) Find the marginal density of X**

* The marginal density of X, denoted by f<sub>X</sub>(x), is found by integrating the joint density function f(x, y) with respect to y:

   f<sub>X</sub>(x) = ∫<sub>-∞</sub><sup>∞</sup> f(x, y) dy 

   Since f(x, y) = e<sup>-x-y</sup> for x > 0 and y > 0, and 0 otherwise, we have:

   f<sub>X</sub>(x) = ∫<sub>0</sub><sup>∞</sup> e<sup>-x-y</sup> dy 

   = e<sup>-x</sup> ∫<sub>0</sub><sup>∞</sup> e<sup>-y</sup> dy 

   = e<sup>-x</sup> [-e<sup>-y</sup>]<sub>0</sub><sup>∞</sup> 

   = e<sup>-x</sup> (0 - (-1)) 

   = e<sup>-x</sup> for x > 0 

   and 0 otherwise.

**b) Find the conditional density of Y given X = x**

* The conditional density of Y given X = x, denoted by f<sub>Y|X</sub>(y|x), is given by:

   f<sub>Y|X</sub>(y|x) = f(x, y) / f<sub>X</sub>(x) 

   Since we found f<sub>X</sub>(x) = e<sup>-x</sup>, we have:

   f<sub>Y|X</sub>(y|x) = (e<sup>-x-y</sup>) / (e<sup>-x</sup>) 

   = e<sup>-y</sup> for y > 0 

   and 0 otherwise.

**c) Find the first joint moment of X and Y**

* The first joint moment of X and Y, denoted by E[XY], is given by:

   E[XY] = ∫<sub>-∞</sub><sup>∞</sup> ∫<sub>-∞</sub><sup>∞</sup> x*y*f(x, y) dx dy

   = ∫<sub>0</sub><sup>∞</sup> ∫<sub>0</sub><sup>∞</sup> x*y*e<sup>-x-y</sup> dx dy

   = ∫<sub>0</sub><sup>∞</sup> x*e<sup>-x</sup> [∫<sub>0</sub><sup>∞</sup> y*e<sup>-y</sup> dy] dx

* Note that ∫<sub>0</sub><sup>∞</sup> y*e<sup>-y</sup> dy is the expected value of an exponential random variable with parameter 1, which is equal to 1.

   Therefore,

   E[XY] = ∫<sub>0</sub><sup>∞</sup> x*e<sup>-x</sup> dx

* This integral also represents the expected value of an exponential random variable with parameter 1, which is equal to 1.

   **So, the first joint moment of X and Y is E[XY] = 1.**

**In summary:**

* The marginal density of X is f<sub>X</sub>(x) = e<sup>-x</sup> for x > 0.
* The conditional density of Y given X = x is f<sub>Y|X</sub>(y|x) = e<sup>-y</sup> for y > 0.
* The first joint moment of X and Y is E[XY] = 1.